Subgroups — #26106945

Order 12032.

574 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
2	#20243134	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×D₄⋊(C₄₇⋊C₈)	C₂	1	2	yes	maximal
4	#20241229	C₂	1	2	yes	maximal, complemented
5	C₄₇×#2293	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄×C₄₇⋊C₈	—	2	1	no	—
8	C₂×C₄×C₄₇⋊C₈	C₂²	1	2	yes	—
9	#15688878	—	2	1	no	—
10	#15688878	—	2	2	no	—
11	D₄⋊(C₄₇⋊C₈)	C₄	1	8	yes	complemented
12	C₂×C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×#117×C₄₇	—	2	1	no	—
14	C₄₇×#390	C₄	1	2	yes	—
15	C₄₇×#389	C₄	1	2	yes	—
16	C₄₇×#388	—	2	2	no	—
17	C₄×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₄×C₂²⋊C₄×C₄₇	—	2	1	no	—
19	C₄³×C₄₇	C₂²	1	1	yes	characteristic
20	C₂²×C₄₇⋊C₈	—	2	2	no	—
21	C₂²×C₄₇⋊C₈	—	2	2	no	—
22	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₄₇⋊C₈)	C₂×C₄	1	4	yes	—
24	C₄×C₄₇⋊C₈	—	2	2	no	—
25	C₄×C₄₇⋊C₈	—	4	2	no	—
26	C₄×C₄₇⋊C₈	C₂×C₄	1	4	yes	—
27	C₂²×D₄×C₄₇	C₂×C₄	1	1	yes	characteristic
28	C₂³×C₄×C₄₇	—	2	1	no	—
29	C₄×D₄×C₄₇	—	2	2	no	—
30	C₄×D₄×C₄₇	C₂×C₄	1	4	yes	—
31	#117×C₄₇	—	4	2	no	—
32	#117×C₄₇	—	4	2	no	—
33	C₂×C₄⋊C₄×C₄₇	—	2	2	no	—
34	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
35	C₂×C₄⋊C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
37	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
38	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
39	C₂×C₄²×C₄₇	—	2	1	no	—
40	C₂×C₄²×C₄₇	C₂×C₄	1	2	yes	—
41	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
42	C₂×C₄²×C₄₇	D₄	1	2	yes	—
43	C₂×C₄²×C₄₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
45	#95×C₄₇	—	2	2	no	—
46	#95×C₄₇	—	2	1	no	—
47	C₂⁴×C₄₇	—	2	1	no	—
48	C₂×D₄×C₄₇	—	2	1	no	—
49	C₂×D₄×C₄₇	—	2	4	no	—
50	C₂×D₄×C₄₇	C₄²	1	2	yes	—
51	C₂²×C₄×C₄₇	—	2	1	no	—
52	C₂²×C₄×C₄₇	—	2	4	no	—
53	C₂²×C₄×C₄₇	—	2	2	no	—
54	C₂²×C₄×C₄₇	—	4	1	no	—
55	C₂²×C₄×C₄₇	—	4	1	no	—
56	C₂²×C₄×C₄₇	—	4	2	no	—
57	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
62	C₄⋊C₄×C₄₇	—	2	1	no	—
63	C₄⋊C₄×C₄₇	—	2	4	no	—
64	C₄⋊C₄×C₄₇	—	4	4	no	—
65	C₄⋊C₄×C₄₇	C₄²	1	2	yes	—
66	C₂²⋊C₄×C₄₇	—	2	4	no	—
67	C₂²⋊C₄×C₄₇	—	4	4	no	—
68	C₂²⋊C₄×C₄₇	—	4	4	no	—
69	C₄²×C₄₇	—	2	4	no	—
70	C₄²×C₄₇	—	2	2	no	—
71	C₄²×C₄₇	—	4	2	no	—
72	C₄²×C₄₇	—	4	1	no	—
73	C₄²×C₄₇	—	4	1	no	—
74	C₄²×C₄₇	D₈	1	2	yes	—
75	C₄²×C₄₇	SD₁₆	1	2	yes	—
76	C₄²×C₄₇	C₂²×C₄	1	2	yes	—
77	C₄²×C₄₇	#41	1	2	yes	—
78	C₂×C₄₇⋊C₈	—	2	4	no	—
79	C₂×C₄₇⋊C₈	—	2	4	no	—
80	C₂×C₄₇⋊C₈	—	4	4	no	—
81	C₂×C₄₇⋊C₈	—	4	2	no	—
82	C₂×C₄₇⋊C₈	—	4	2	no	—
83	C₂³×C₄₇	—	2	2	no	—
84	C₂³×C₄₇	—	4	2	no	—
85	C₂³×C₄₇	—	4	1	no	—
86	C₂³×C₄₇	—	4	1	no	—
87	C₂³×C₄₇	—	4	2	no	—
88	C₂³×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	D₄×C₄₇	—	2	4	no	—
90	D₄×C₄₇	—	4	2	no	—
91	C₂×C₄×C₄₇	—	2	2	no	—
92	C₂×C₄×C₄₇	—	2	2	no	—
93	C₂×C₄×C₄₇	—	2	2	no	—
94	C₂×C₄×C₄₇	—	2	2	no	—
95	C₂×C₄×C₄₇	—	2	2	no	—
96	C₂×C₄×C₄₇	—	2	2	no	—
97	C₂×C₄×C₄₇	—	2	2	no	—
98	C₂×C₄×C₄₇	—	4	1	no	—
99	C₂×C₄×C₄₇	—	4	1	no	—
100	C₂×C₄×C₄₇	—	4	4	no	—
101	C₂×C₄×C₄₇	—	4	4	no	—
102	C₂×C₄×C₄₇	—	4	2	no	—
103	C₂×C₄×C₄₇	—	4	2	no	—
104	C₂×C₄×C₄₇	—	4	2	no	—
105	C₂×C₄×C₄₇	#102	1	4	yes	—
106	C₂×C₄×C₄₇	C₂×C₄²	1	1	yes	characteristic
107	C₂×C₄×C₄₇	#117	1	1	yes	characteristic
108	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
109	C₂×C₄×C₄₇	C₄×D₄	1	2	yes	—
110	C₂×C₄×C₄₇	#131	1	4	yes	—
111	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
113	C₂×C₄×C₄₇	#135	1	2	yes	—
114	C₄₇⋊C₈	—	4	4	no	—
115	C₄₇⋊C₈	—	4	4	no	—
116	#8183	—	47	1	no	maximal, Sylow
117	C₂²×C₄₇	—	4	2	no	—
118	C₂²×C₄₇	—	4	2	no	—
119	C₂²×C₄₇	—	4	2	no	—
120	C₂²×C₄₇	—	8	2	no	—
121	C₂²×C₄₇	—	8	2	no	—
122	C₂²×C₄₇	C₄×C₂²⋊C₄	1	1	yes	characteristic
123	C₂²×C₄₇	#408	1	2	yes	—
124	C₂²×C₄₇	C₂×#102	1	1	yes	characteristic
125	C₂²×C₄₇	#416	1	1	yes	characteristic
126	C₂²×C₄₇	#421	1	2	yes	—
127	C₄×C₄₇	—	2	4	no	—
128	C₄×C₄₇	—	2	2	no	—
129	C₄×C₄₇	—	4	2	no	—
130	C₄×C₄₇	—	4	2	no	—
131	C₄×C₄₇	—	8	2	no	—
132	C₄×C₄₇	#405	1	2	yes	—
133	C₄×C₄₇	#433	1	2	yes	—
134	C₄×C₄₇	C₄×D₈	1	2	yes	—
135	C₄×C₄₇	C₄×SD₁₆	1	2	yes	—
136	#2293	D₄₇	1	1	yes	characteristic
137	C₄×C₄⋊C₈	—	47	1	no	—
138	#1744	—	47	1	no	—
139	C₂×D₄⋊C₈	—	47	2	no	—
140	#1279	—	47	2	no	—
141	C₂×C₄₇	—	8	2	no	—
142	C₂×C₄₇	#1480	1	2	yes	—
143	C₂×C₄₇	#1482	1	2	yes	—
144	C₂×C₄₇	#1750	1	1	yes	characteristic
145	C₂×C₄₇	#1759	1	1	yes	characteristic
146	C₂×C₄₇	C₄×#102	1	1	yes	characteristic
147	C₂×C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
148	C₂×#117	—	2	1	no	—
149	C₂×C₄⋊C₈	—	47	2	no	—
150	C₂×C₄×C₈	—	47	2	no	—
151	C₂×C₄×C₈	—	94	1	no	—
152	#390	Dic₄₇	1	2	yes	—
153	#389	Dic₄₇	1	2	yes	—
154	#388	—	2	2	no	—
155	C₄×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
156	C₄×C₂²⋊C₄	—	2	1	no	—
157	C₄³	C₂×D₄₇	1	1	yes	characteristic
158	#336	—	94	2	no	—
159	#336	—	94	1	no	—
160	D₄⋊C₈	—	47	8	no	—
161	C₄₇	#8183	1	1	yes	characteristic, complemented, Sylow
162	C₂²×D₄	C₂×Dic₄₇	1	1	yes	characteristic
163	C₂³×C₄	—	2	1	no	—
164	C₂²×C₈	—	94	2	no	—
165	C₂²×C₈	—	94	2	no	—
166	C₄×D₄	—	2	2	no	—
167	C₄×D₄	C₄×D₄₇	1	4	yes	—
168	#117	—	4	2	no	—
169	#117	—	4	2	no	—
170	C₂×C₄⋊C₄	—	2	2	no	—
171	C₂×C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
172	C₂×C₄⋊C₄	C₂×Dic₄₇	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	1	no	—
174	C₂×C₂²⋊C₄	—	2	2	no	—
175	C₂×C₂²⋊C₄	—	2	2	no	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	C₂×Dic₄₇	1	2	yes	—
178	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
179	C₂×C₄²	C₂²⋊D₄₇	1	2	yes	—
180	C₂×C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
181	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
182	C₄⋊C₈	—	47	4	no	—
183	C₄⋊C₈	—	94	2	no	—
184	C₄×C₈	—	47	4	no	—
185	C₄×C₈	—	94	2	no	—
186	C₄×C₈	—	188	2	no	—
187	#95	—	2	2	no	—
188	#95	—	2	1	no	—
189	C₂⁴	—	2	1	no	—
190	C₂×D₄	—	2	1	no	—
191	C₂×D₄	—	2	4	no	—
192	C₂×D₄	C₄×Dic₄₇	1	2	yes	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	4	1	no	—
197	C₂²×C₄	—	4	1	no	—
198	C₂²×C₄	—	4	2	no	—
199	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
200	C₂²×C₄	C₂²⋊Dic₄₇	1	2	yes	—
201	C₂²×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
202	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
203	C₂²×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
204	C₂×C₈	—	94	4	no	—
205	C₂×C₈	—	94	4	no	—
206	C₂×C₈	—	188	4	no	—
207	C₂×C₈	—	188	2	no	—
208	C₂×C₈	—	188	2	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	4	no	—
212	C₄⋊C₄	C₄×Dic₄₇	1	2	yes	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	4	4	no	—
215	C₂²⋊C₄	—	4	4	no	—
216	C₄²	—	2	4	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	4	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	#10625334	1	2	yes	—
222	C₄²	#10625335	1	2	yes	—
223	C₄²	C₂×C₄×D₄₇	1	2	yes	—
224	C₄²	#10625350	1	2	yes	—
225	C₂³	—	2	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	1	no	—
228	C₂³	—	4	1	no	—
229	C₂³	—	4	2	no	—
230	C₂³	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
231	D₄	—	2	4	no	—
232	D₄	—	4	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	1	no	—
242	C₂×C₄	—	4	4	no	—
243	C₂×C₄	—	4	4	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	—	4	2	no	—
246	C₂×C₄	—	4	2	no	—
247	C₂×C₄	#13093945	1	4	yes	—
248	C₂×C₄	C₂×C₄×Dic₄₇	1	1	yes	characteristic
249	C₂×C₄	#13093983	1	1	yes	characteristic
250	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
251	C₂×C₄	C₄×C₂²⋊D₄₇	1	2	yes	—
252	C₂×C₄	#13094031	1	4	yes	—
253	C₂×C₄	C₂×#10625334	1	1	yes	characteristic
254	C₂×C₄	C₂×#10625335	1	1	yes	characteristic
255	C₂×C₄	#13094044	1	2	yes	—
256	C₈	—	188	4	no	—
257	C₈	—	188	4	no	—
258	C₂²	—	4	2	no	—
259	C₂²	—	4	2	no	—
260	C₂²	—	4	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	—	8	2	no	—
263	C₂²	C₄×C₂²⋊Dic₄₇	1	1	yes	characteristic
264	C₂²	#15689090	1	2	yes	—
265	C₂²	C₂×#13093945	1	1	yes	characteristic
266	C₂²	#15689119	1	1	yes	characteristic
267	C₂²	#15689140	1	2	yes	—
268	C₄	—	2	4	no	—
269	C₄	—	2	2	no	—
270	C₄	—	4	2	no	—
271	C₄	—	4	2	no	—
272	C₄	—	8	2	no	—
273	C₄	#15689079	1	2	yes	—
274	C₄	#15689184	1	2	yes	—
275	C₄	C₄×#10625334	1	2	yes	—
276	C₄	C₄×#10625335	1	2	yes	—
277	C₂	—	8	2	no	—
278	C₂	#20241787	1	2	yes	—
279	C₂	#20241796	1	2	yes	—
280	C₂	#20243154	1	1	yes	characteristic
281	C₂	#20243198	1	1	yes	characteristic
282	C₂	C₄×#13093945	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#2293	C₂
socle	C₂³×C₄₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₄×C₄₇	C₂×C₄
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#2293	D₄₇
hypocenter	C₄₇	#8183
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #2293

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₄×C₄⋊(C₄₇⋊C₈)	C₂	semidirect
2. #20243134	C₂	semidirect
4. #20241229	C₂	semidirect
11. D₄⋊(C₄₇⋊C₈)	C₄	semidirect
161. C₄₇	#8183	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₄×C₄⋊(C₄₇⋊C₈)	2
#20243134	2
C₂×D₄⋊(C₄₇⋊C₈)	2
#20241229	2
C₄₇×#2293	2
#8183	47